True or False
(a) If X ∩ Y = ∅ then the two events X and Y are independent?
(b) If event X is independent of event Y, then X^c is independent of Y?
(c) For a discrete random variable X, we have limx->∞ pX(x) = 0?
(d) For a continuous random variable X, we have limx->∞ fX(x) = 0?
(e) For a continuous random variable X, we have limx->0 fX(x) ≤ 1?
(f) For two discrete random variables X and Y , we have ∑x pX,Y (x, y) = 1?
(g) For two discrete random variables X and Y , we have ∑x pX|Y (x|y) = 1?
(h) If random variable A is continuous, then for a real-valued function g, Y = g(A) is a continuous random variable?
(i) If random variable A is discrete, then for a real-valued function g, Y = g(A) is a discrete random variable?
(j) For a random variable A, {A = 0} ∩ {A = 1} = ∅
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6. Determine whether each of the following is true or false (note: the statement is true if it is always true, otherwise it is false). If you say it is true then refer to a known result or give a proof, while if you say it is false then give a counterexample, i.e., a particular case where it fails. (a) If A, B and C are independent, the Pr(AlBnc)- Pr (A) (b) The events S., A are independent (S is...
True or False With explanation please. e. If two events are mutually exclusive, they are àl f. The mean of a discrete random variable X is penide found by multiplying each possible value of X by its own probability and then adding all the products together; that is XPO g. Two events A and B are said to be independent if P(A and B)y- PIA)-P(B) h. Assume that X is a normally distributed random variable with a mean ofμ and...
2. Let X and Y be two independent discrete random variables with the probability mass functions PX- = i) = (e-1)e-i and P(Y = j-11' for i,j = 1, 2, Let {Uni2 1} of i.i.d. uniform random variables on [0, 1]. Assume the sequence {U i independent of X and Y. Define M-max(UhUn Ud. Find the distribution
[1] The joint probability density function of two continuous random variables X and Y is fx,x(x, y) = {6. sc, 0 <y s 2.y = x < 4-y otherwise Find the value of c and the correlation of X and Y.
TRUE OR FALSE _______23. Events are independent when the occurrence of one event has no effect on the probability that another will occur. _______24.The P(x) is always 0 ≤ P(x) ≤ 1. _______25. The mean of the discrete probability distribution for a discrete random variable is called its expected value
True or False With explanation please. 1- True or false: a. If A is an event of a sample space with P(A)-P(AS), then P(A)-0.5 b. Under certain conditions, it is possible that the sum of the probabilities of all the sample points in a sample space is less than one P(A or B)-P(A)+P(B) P(A and B) P(A).P(B) by its own probability and then adding all the products together; that is P deviation of σ. If x is converted to the...
We said in class that two events A and B are indep(ndent if μ(An B) 6. μ(A)a(B). Sinilarly, two random variables X and Y are said to be independent if their joint density fx.y(r,y) can be expressed as the product of the marginal densities fx(x)fv(y). Let X and Y be independent (scalar) random variables, and ZX Y be a new random variable defined as the sum of X and Y. Show that the moment generating function mz(t) of Z is...
Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events (think of incoming phone calls, customers entering a bank, car accidents, etc.) that occur up to and including time t where the occurrence times of these events satisfy the following three conditions Events only occur after time 0, i.e., X(t)0 for t0 If N (1, 2], where 0< t t2, denotes the number of events that occur in the time interval (t1,...
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
True or False With explanation please. f. If a and b are constants and X is a randoin variable , then,ar(aX +b)-apa (X) t l' g. Tickets numbered from 2 to 11 are mixed up and then a ticke is draw X 1 n at random. The probability that the ticket drawn has prime number is 2/11. N If the covariance of two random variables is pasitive and the first number is negative then most probably the secoud number will...